Note that since $\rm a^2=e\iff a=a^{-1}$, the map $\rm\Phi_a: x\mapsto axa=axa^{-1}$ is conjugation, which is a special type of automorphism. In particular $\rm\Phi_a(xy)=\Phi_a(x)\Phi_a(y)$ and thus $\rm\Phi_a(x^{n})=\Phi_a(x)^n$ for any elements $\rm x,y$ and integer ${\rm n}\in{\bf Z}$. We employed the latter exponential distributivity above.